Problem 61 - Discrete Mathematics with Applications

Question

Suppose we introduce a mixed pair of -month-old rabbits into a large enclosure on the first day of a certain month. By the end of each month, the rabbits become mature and each pair produces \(k-1\) mixed pairs of offspring at the beginning of the following month. Wote: \(k \geq 2 .\) ) For instance, at the beginning of the second month, there is one pair of 2 -month-old rabbits and \(k-1\) pairs of 0 -month-olds; at the beginning of the third month, there is one pair of 3-month-olds, \(k-1\) pairs of 1 -month-olds, and \(k(k-1)\) pairs of 0 -month-olds. Assume the rabbits are immortal. Let \(a_{n}\) denote the average age of the rabbit pairs at the beginning of the \(n\) month. (P. Filiponi, 1990 ) Define \(a_{n}\) recursively.

Step-by-Step Solution

Answer

The recursive relationship for the average age of rabbit pairs at the beginning of the n-th month is given by: \[a_{n} = \frac{A(n-1) + (n-1) r(n-1)}{t(n)}\] where \(A(n)\) is the total age of rabbit pairs at the beginning of the n-th month, \(r(n)\) is the number of new rabbit pairs at the beginning of the n-th month, and \(t(n)\) is the total number of rabbit pairs at the beginning of the n-th month.

Step 1: Determine the number of new rabbit pairs for each month

In order to find the average age of the rabbit pairs, we first need to find out how many new rabbit pairs there are each month. We are given that the initial pair of rabbits produce \(k-1\) mixed pairs of offspring at the beginning of each month. At the beginning of the second month, there are \(k-1\) pairs of 0-month-olds. At the beginning of the third month, there are \(k(k-1)\) pairs of 0-month-olds. This is because each pair of 1-month-olds produce \(k-1\) pairs of 0-month-olds. To generalize this, we can express the number of new rabbit pairs at the beginning of the n-th month as a function of \(k\) and \(n\). Let's denote this as \(r(n)\): \[ r(n) = (k-1)^{n-1} \]

Step 2: Calculate the total number of rabbit pairs at the beginning of each month

Now, we need to find the total number of rabbit pairs at the beginning of each month. At the start of month 1, there is 1 pair of rabbits. At the start of month 2, there is 1 pair of 2-month-olds and \(k-1\) pairs of 0-month-olds, making a total of \(1 + (k-1)\) rabbit pairs. At the start of month 3, we have: - 1 pair of 3-month-olds - \(k-1\) pairs of 1-month-olds - \(k(k-1)\) pairs of 0-month-olds Total number of rabbit pairs at the start of month 3 is: \(1 + (k-1) + k(k-1)\). We can generalize this to find the total number of rabbit pairs at the beginning of the n-th month, denoted as \(t(n)\): \[ t(n) = \sum_{i=1}^{n} r(i) = 1 + (k-1)(1 + k-1 + (k-1)^2 + \cdots + (k-1)^{n-2}) \]

Step 3: Calculate the total age of rabbit pairs at the beginning of each month

Now, let's find the total age of rabbit pairs at the beginning of each month. We denote this as \(A(n)\). At the beginning of month 2: \[ A(2) = 2*(1) + 0*(k-1) = 2 \] At the beginning of month 3: \[ A(3) = 3*(1) + 1*(k-1) + 0*(k(k-1)) = 3 + k - 1 \] To find the total age of rabbit pairs at the beginning of the n-th month, we can use the following recursive formula: \[ A(n) = A(n-1) + (n-1) r(n-1) \]

Step 4: Define the average age of rabbit pairs at the beginning of each month

Now, we can define the average age of rabbit pairs at the beginning of the n-th month, denoted as \(a_{n}\). \[a_{n} = \frac{A(n)}{t(n)}\]

Step 5: Define the recursive relationship for the average age

Using the formulas for \(A(n)\), \(t(n)\), and \(a_{n}\), we can define a recursive relationship for the average age of rabbit pairs at the beginning of the n-th month: \[a_{n} = \frac{A(n)}{t(n)} = \frac{A(n-1) + (n-1) r(n-1)}{t(n)}\] Now we have a recursive expression for the average age of rabbit pairs at the beginning of the n-th month based on the population growth and rabbit pair ages.

Key Concepts

Fibonacci sequence

The Fibonacci sequence is a fascinating pattern of numbers that appears often in the natural world. It begins with two initial numbers, typically 0 and 1, and each subsequent number is the sum of the two preceding numbers. The sequence looks like this: 0, 1, 1, 2, 3, 5, 8, and so on. The Fibonacci sequence is a simple yet powerful example of a recurrence relation, where each term is defined as a function of its previous terms.
The Fibonacci numbers have various applications:

• Mathematics: These numbers are used in computational functions like recursive algorithms and numerical patterns.
• Biology: The sequence is observed in the arrangement of leaves on a stem and the fruit sprouts of a pineapple.
• Art: The golden ratio derived from the sequence is often used in art and architecture to achieve aesthetic balance.

In the exercise, the concept of generating numbers using a recursive relationship is similar to the Fibonacci sequence's nature, where a pattern or number series is built from previous terms.

population growth models

Population growth models describe how populations change over time under various assumptions. These models are essential for understanding and predicting how animal colonies expand and contract. In essence, population growth models use mathematical equations to estimate future population sizes based on current conditions. There are different types of population growth models:

• Exponential Growth Model: Suitable for populations growing without constraints, assuming unlimited resources.
• Logistic Growth Model: Takes into consideration factors like limited resources, showing a slowing growth rate as population nears carrying capacity.
• Recurrence Relation Models: As seen in the exercise, these rely on recursive equations to update a population based on defined rules.

The rabbit population growth problem in the exercise utilizes a recurrence relation model, demonstrating how mathematical equations can model the incremental growth of rabbit pairs month by month, considering the continuous birth of offspring each month.

average age calculation

Calculating the average age of a group is a common problem across various fields. The average age provides insight into the temporal dynamics of a population, helping in understanding aspects like lifecycle stages and population maturity. The exercise problem involves calculating the average age of rabbit pairs, which leverages several key concepts:

• Sum of Ages: It calculates the total sum of ages of all individuals to determine the overall age count.
• Total Number of Individuals: Knowing the total number of rabbit pairs helps in deriving the average age.
• Recursive Relationships: The exercise shows how to update and calculate these metrics using recursive equations.

The formula for finding the average age, such as \( a_n = \frac{A(n)}{t(n)} \), simplifies the task by dividing the total age by the number of pairs, offering a clear snapshot of the population's age profile as simulations progress over months. This recursive method supports ongoing evaluations as the population structure continuously changes with new births.